Vibrations of viscoelastic isotropic rectangular plates of an auxetic metamaterial are considered in a linear formulation. The problem is described by a linear integro-differential partial differential equation with initial and boundary conditions. The weakly singular relaxation kernel of Koltunov-Rzhanitsyn is used. Using the Bubnov-Galerkin method, the resulting equation is reduced to a linear ordinary integro-differential equation with respect to the time function. This equation is solved by a numerical method based on the use of quadrature formulas, eliminating singularities in the relaxation kernel. The effect on the amplitude-frequency characteristic of vibrations of a viscoelastic isotropic rectangular plate of a metamaterial with a negative Poisson's ratio is studied.